Induced Disjoint Paths in AT-free graphs

Golovach, Petr A.; Paulusma, Daniël; Leeuwen, Erik Jan van

doi:10.1016/j.jcss.2021.10.003

Cited by 6 publications

(3 citation statements)

References 43 publications

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“…6). Namely, Induced Disjoint Paths is linear-time solvable for circular-arc graphs [10]; polynomial-time solvable for chordal graphs [1], AT-free graphs [11], graph classes of bounded mim-width [13]; and NPcomplete for claw-free graphs [6], line graphs of triangle-free chordless graphs [27] and thus for (theta,wheel)-free graphs, and for planar graphs; the last result follows from a result of Lynch [21] (see [11]). Moreover, Induced Disjoint Paths is XP with parameter k for (theta,wheel)-free graphs [27] and even FPT with parameter k for claw-free graphs [9] and planar graphs [15]; the latter can be extended to graph classes of bounded genus [18].…”

Section: Known Resultsmentioning

confidence: 99%

“…Let F be the subgraph of the blob graph G • induced by all connected subsets X in G that have size at most 11 , such that X contains all vertices of one set from Z and no vertices from any other set of Z . Then F has polynomial size, as it has O(n 11 ) vertices, so we can construct F in polynomial time. By Lemma 2, F is P 6 -free.…”

Section: Using the Blob Graph Approachmentioning

confidence: 99%

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Induced Disjoint Paths and Connected Subgraphs for H-Free Graphs

Martin

Paulusma²,

Smith³

et al. 2022

Lecture Notes in Computer Science

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Paths P 1 , . . . , P k in a graph G = (V , E) are mutually induced if any two distinct P i and P j have neither common vertices nor adjacent vertices. The Induced Disjoint Paths problem is to decide if a graph G with k pairs of specified vertices (s i , t i ) contains k mutually induced paths P i such that each P i starts from s i and ends at t i . This is a classical graph problem that is NP-complete even for k = 2. We introduce a natural generalization, Induced Disjoint Connected Subgraphs: instead of connecting pairs of terminals, we must connect sets of terminals. We give almostcomplete dichotomies of the computational complexity of both problems for H -free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph. Finally, we give a complete classification of the complexity of the second problem if the number k of terminal sets is fixed, that is, not part of the input.

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Section: Known Resultsmentioning

confidence: 99%

Section: Using the Blob Graph Approachmentioning

confidence: 99%

Induced Disjoint Paths and Connected Subgraphs for H-Free Graphs

Martin

Paulusma²,

Smith³

et al. 2022

Lecture Notes in Computer Science

Self Cite

View full text Add to dashboard Cite

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“…The LIPP is known to be NP‐hard (Garey and Johnson, 1979). Thus, several techniques have been proposed for tackling the problem, including approximation algorithms (Karger et al., 1997), parameterized complexity (Fiala et al., 2012; Golovach et al., 2012, 2015; Bergougnoux and Kanté, 2021; Golovach et al., 2022), integer programming (Matsypura et al., 2019; Bökler et al., 2020a; Marzo et al., 2022), backtracking (Marzo and Ribeiro, 2021), and heuristics (Matsypura et al., 2019; Marzo and Ribeiro, 2021). It is worth mentioning, however, that there are classes of graphs for which the problem is known to be polynomially solvable (Gavril, 2002; Kratsch et al., 2003; Ishizeki et al., 2008; Jaffke et al., 2020a).…”

Section: Literature Reviewmentioning

confidence: 99%

MIP formulations for induced graph optimization problems: a tutorial

Melo

Ribeiro

2023

Int Trans Operational Res

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Given a graph G=false(V,Efalse)$G=(V,E)$ and a subset of its vertices V′⊆V$V^{\prime }\subseteq V$, the subgraph induced by V′$V^{\prime }$ in G is that with vertex set V′$V^{\prime }$ and edge set E′$E^{\prime }$ formed by all the edges in E linking two vertices in V′$V^{\prime }$. Mixed integer programming (MIP) approaches are among the most successful techniques for solving induced graph optimization problems, that is, those related to obtaining maximum or minimum (weighted or not) induced subgraphs with certain properties. In this tutorial, we provide a literature review of some of these problems. Furthermore, we illustrate the use of MIP formulations and techniques for solving combinatorial optimization problems involving induced graphs. We focus on compact formulations and those with an exponential number of constraints that can be effectively solved using branch‐and‐cut procedures. More specifically, we revisit applications of their use for problems of finding induced forests (which correspond to the complement of feedback vertex sets), trees, paths, as well as quasi‐clique partitionings.

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Few Induced Disjoint Paths for H-Free Graphs

Martin

Paulusma

Smith

et al. 2022

Lecture Notes in Computer Science

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Paths P 1 , . . . , P k in a graph G = (V , E) are mutually induced if any two distinct P i and P j have neither common vertices nor adjacent vertices. For a fixed integer k, the k-Induced Disjoint Paths problem is to decide if a graph G with k pairs of specified vertices (s i , t i ) contains k mutually induced paths P i such that each P i starts from s i and ends at t i . Whereas the non-induced version is well-known to be polynomial-time solvable for every fixed integer k, a classical result from the literature states that even 2-Induced Disjoint Paths is NP-complete. We prove new complexity results for k-Induced Disjoint Paths if the input is restricted to H-free graphs, that is, graphs without a fixed graph H as an induced subgraph. We compare our results with a complexity dichotomy for Induced Disjoint Paths, the variant where k is part of the input.

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Induced Disjoint Paths in AT-free graphs

Cited by 6 publications

References 43 publications

Induced Disjoint Paths and Connected Subgraphs for H-Free Graphs

Induced Disjoint Paths and Connected Subgraphs for H-Free Graphs

MIP formulations for induced graph optimization problems: a tutorial

Few Induced Disjoint Paths for H-Free Graphs

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